Caterina Stoppato

Regular functions of a quaternionic variable

Introduction.- 1.Definitions and Basic Results.- 2.Regular Power Series.- 3.Zeros.- 4.Infinite Products.- 5.Singularities.- 6.Integral Representations.- 7.Maximum Modulus Theorem and Applications.- 8.Spherical Series and Differential.- 9.Fractional Transformations and the Unit Ball.- 10.Generalizations and Applications.- Bibliography.- Index.

The Open Mapping Theorem for Regular Quaternionic Functions

The basic results of a new theory of regular functions of a quaternionic variable have been recently stated, following an idea of Cullen. In this paper we prove the minimum modulus principle and the open mapping theorem for regular functions. The proofs involve some peculiar geometric properties of such functions which are of independent interest.

Twistor transforms of quaternionic functions and orthogonal complex structures

The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega\ of R^4. When \Omega\ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega\ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space CP^3.

Poles of regular quaternionic functions

This article studies the singularities of functions of one quaternionic variable which are regular in the sense of Gentili and Struppa (A new theory of regular functions of a quaternionic variable, Adv. Math. 216 (2007), pp. 279–301). The quaternionic Laurent series prove to be regular. The singularities of regular functions are thus classified as removable, essential or poles. The quaternionic analogues of meromorphic complex functions, called semiregular functions, turn out to be quotients of regular functions with respect to an appropriate division operation. This allows a detailed study of the poles and their distribution.

A new series expansion for slice regular functions

Abstract A promising theory of quaternion-valued functions of one quaternionic variable, now called slice regular functions, has been introduced by Gentili and Struppa in 2006. The basic examples of slice regular functions are the power series of type ∑ n ∈ N q n a n on their balls of convergence B ( 0 , R ) = { q ∈ H : | q | R } . Conversely, if f is a slice regular function on a domain Ω ⊆ H then it admits at each point q 0 ∈ Ω an expansion of type f ( q ) = ∑ n ∈ N ( q − q 0 ) ∗ n a n where ( q − q 0 ) ∗ n denotes the n th power of q − q 0 with respect to an appropriately defined multiplication ∗ . However, the information provided by such an expansion is somewhat limited by a fact: if q 0 does not lie on the real axis then the set of convergence of the series in the previous equation needs not be a Euclidean neighborhood of q 0 . We are now able to construct a new type of expansion that is not affected by this phenomenon: an expansion into series of polynomials valid in open subsets of the domain. Along with this construction, we present applications to the computation of the multiplicities of zeros and of partial derivatives.

Singularities of slice regular functions

Beginning in 2006, G. Gentili and D. C. Struppa developed a theory of regular quaternionic functions with properties that recall classical results in complex analysis. For instance, in each Euclidean ball B(0, R) centered at 0 the set of regular functions coincides with that of quaternionic power series converging in B(0, R). In 2009 the author proposed a classification of singularities of regular functions as removable, essential or as poles and studied poles by constructing the ring of quotients. In that article, not only the statements, but also the proving techniques were confined to the special case of balls B(0, R). Quite recently, F. Colombo, G. Gentili and I. Sabadini (2010) and the same authors in collaboration with D. C. Struppa (2009) identified a larger class of domains, on which the theory of regular functions is natural and not limited to quaternionic power series. The present article studies singularities in this new context, beginning with the construction of the ring of quotients and of Laurent-type expansions at points p other than the origin. These expansions, which differ significantly from their complex analogs, allow a classification of singularities that is consistent with the one given in 2009. Poles are studied, as well as essential singularities, for which a version of the Casorati-Weierstrass Theorem is proven.

The Schwarz-Pick lemma for slice regular functions

The celebrated Schwarz-Pick lemma for the complex unit disk is the basis for the study of hyperbolic geometry in one and in several complex variables. In the present paper, we turn our attention to the quaternionic unit ball B. We prove a version of the Schwarz-Pick lemma for self-maps of B that are slice regular, according to the definition of Gentili and Struppa. The lemma has interesting applications in the fixed-point case, and it generalizes to the case of vanishing higher order derivatives.

Recent Developments for Regular Functions of a Hypercomplex Variable

In this paper we survey a series of recent developments in the theory of functions of a hypercomplex variable. The central idea underlying these developments consists in requiring a function to be holomorphic on suitable slices of the space on which the function itself is defined. Specifically, we apply this approach to functions defined on the space ℍ of quaternions, on the space O of octonions, and finally on the Clifford algebra of type (0,3), denoted Cl (0,3). The properties of these functions resemble those of holomorphic functions, and yet the different nature of the three algebras on which we work introduces new and exciting phenomena.

The Zero Sets of Slice Regular Functions and the Open Mapping Theorem

A new class of regular quaternionic functions, defined by power series in a natural fashion, has been introduced in [11, 12]. Several results of the theory recall the classical complex analysis, whereas other results reflect the peculiarity of the quaternionic structure. The recent work [1, 2] identified a larger class of domains, on which the study of regular functions is most natural and not limited to the study of quaternionic power series. In the present paper we extend some basic results concerning the algebraic and topological properties of the zero set to regular functions defined on these domains. We then use these results to prove the Maximum and Minimum Modulus Principles and a version of the Open Mapping Theorem in this new setting.

The algebra of slice functions

In this paper we study some fundamental algebraic properties of slice functions and slice regular functions over an alternative